www.imstat.org/aihp 2012, Vol. 48, No. 2, 579–608
DOI:10.1214/11-AIHP417
© Association des Publications de l’Institut Henri Poincaré, 2012
The unscaled paths of branching Brownian motion
Simon C. Harris
aand Matthew I. Roberts
baDepartment of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, United Kingdom. E-mail:S.C.Harris@bath.ac.uk bLaboratoire de Probabilités et Modèles Aléatoires, Université Paris VI, 175 rue du Chevaleret, 75013 Paris, France.
E-mail:mattiroberts@gmail.com
Received 23 April 2010; revised 20 January 2011; accepted 31 January 2011
Abstract. For a setA⊂C[0,∞), we give new results on the growth of the number of particles in a branching Brownian motion whose paths fall withinA. We show that it is possible to work without rescaling the paths. We give large deviations probabilities as well as a more sophisticated proof of a result on growth in the number of particles along certain sets of paths. Our results reveal that the number of particles can oscillate dramatically. We also obtain new results on the number of particles near the frontier of the model. The methods used are entirely probabilistic.
Résumé. Considérons un mouvement Brownien branchant. Nous nous intéressons au nombre de particules dont le chemin reste dans un ensemble fixéA⊂C[0,∞). Nous montrons qu’il n’est pas nécessaire de renormaliser les chemins. Nous donnons les probabilités de grandes déviations, ainsi qu’une preuve plus sophistiquée pour un résultat concernant la croissance du nombre de particules dans certains ensembles. Nos résultats démontrent que ce nombre de particules peut fortement osciller. Nous obtenons aussi des résultats nouveaux concernant le nombre de particules proches de la frontière du système. Nos méthodes sont purement probabilistes.
MSC:60J80
Keywords:Branching Brownian motion; Large deviations; Survival probability; Law of large numbers
1. Introduction
One of the most natural questions to ask about branching Brownian motion (BBM) concerns the position of the extremal particle – the particle with maximal position at each timet≥0. It is well-known that its speed – its position divided by time – converges almost surely to√
2rast→ ∞, whereris the branching rate of the system. In fact, far more precise results are available, such as that given by Bramson [1] via some powerful and explicit analysis of the Brownian bridge.
Once we know the speed of the extremal particle at large timest, we might ask about its history: have its ancestors stayed close to the critical speed throughout, or have they hovered around in the mass of particles near the origin and made a late dash as we get close to timet? One way of interpreting this question is to consider branching Brownian motion with absorption. One imagines an absorbing lineL(t )= −x+γ twhereγ is a constant close to the critical value√
2r, such that whenever a particle hits the lineL(t )it disappears and is removed from the system. Are there any particles still present at large times? If so then we may consider them to have stayed “close” to the extremal edge of the system.
This model for BBM with killing on the line was studied by Kesten [10], who discovered asymptotics for extinction probabilities and numbers of particles in intervals of the area above the absorbing line. To choose two examples of particular interest, Kesten shows that ifγ <√
2r then there is strictly positive probability thatN (t )never becomes
empty; and that in the critical case γ =√
2r, the probability that there is at least one particle present at time t is approximately exp(−kt1/3)for some positive constantk. Thus it is possible for particles to stay above the line Γ (t)= −x+γ tfor all time wheneverγ <√
2r, and that this is not the case whenγ=√
2r. Our next question might be: can particles stay withintβ (plus a constant, say) of the critical line forβ∈(0,1)? Indeed, we could also attempt to generalise by moving away from the critical line – given a pathf:[0,∞)→R, are there particles that stay close tof and, if so, how close? Such questions provide motivation for this article.
The classicalscaled path properties of branching Brownian motion (BBM) have now been well-studied: for ex- ample, see Lee [13] and Hardy and Harris [3] for large deviation results on “difficult” paths which have a small probability of any particle following them, and Git [2] and Harris and Roberts [6] for the almost sure growth rate of the number of particles near “easy” paths along which we see exponential growth in the number of particles. To give these results, the paths of a BBM are rescaled onto the interval[0,1], echoing the approach of Schilder’s theorem for a single Brownian motion.
In this article, as suggested above, we consider a problem similar in theme, but from a more naive viewpoint.
We are given a fixed set of paths A⊂C[0,∞)and we want to know how many particles in a BBM have paths within this setA. Similar problems in the case of a single Brownian motion have been considered by Kesten [10] and Novikov [17]. The simplest case is to consider the ballB(f, L)of fixed widthL >0 about a single continuous path f:[0,∞)→R, (we will, however, consider more general sets of paths). Clearly there is a positive probability that no particle will stay within this fixed “tube” – indeed, the very first particle could wander away fromf before it has the chance to give birth to another – and in this event we say that the process becomes extinct.
The intuition is that the growth of the population due to branching is in constant competition with the “deaths” due to particles failing to follow the functionf. Thus a natural condition arises: if the gradient off is too large, then the process eventually dies out almost surely and we may ask for the large deviation probabilities of survival up to large times; otherwise, if the gradient off remains sufficiently small, then we may condition on non-extinction and give an almost sure result on the number of particles along the path.
One payoff for our less classical approach is that we immediately see a dramatic oscillation in the number of particles along certain paths. This unusual behaviour (not seen in the existing literature) has a simple explanation which we demonstrate via some illuminating examples in Section3.
In our proofs, we take advantage of spine techniques to interpret the change of measure given by a carefully chosen martingale. The spine tools give us an intuitive probabilistic handle on the problem, without which we would certainly need substantial extra technical work in several areas. Our particular change of measure involves forcing one particle (the spine) to stay within a tube of varying radiusL(t ),t≥0 about a functionf. This change of measure is the result of a new martingale which we develop in Section4. We then use the spine decomposition first introduced by Lyons et al. [15], which allows us to bound the growth of the system by looking at the births along the spine.
Even with the spine theory the problem retains significant difficulty inherent in its time-inhomogeneity. This fact is underlined by the observation that even in the caseA=B(f, L)we are essentially considering a one-dimensional branching diffusion with time-dependent drift, and asking how many particles remain within a bounded domain about the origin. It turns out that the main difficulty is in showing that extinction of the process coincides (to within a null set) with the event that the limit of our martingale is zero. Standard tools – analytic or probabilistic – cannot be applied;
instead we proceed by our own methods in Section6, using in particular an identity from Harris and Roberts [7].
For simplicity, we consider only standard one-dimensional binary branching Brownian motion, but we note that our work could be extended to a wide range of other branching diffusions. In particular the spine methods are well-suited to the situation where each particle gives birth to a random number of new particles, and methods similar to those used in the original papers of Lyons et al. [11,14,15] could be used to extend our result.
Our main theorem concerns only sets of paths away from criticality. However, by adapting the methods from the proof of this theorem, we are able to obtain new results on the number of particles near the extremes of the system (see Theorems2.3and2.4). These results answer the questions raised in the above discussion and, as was mentioned there, should be compared to the work of Bramson [1] on the position of the right-most particle, and of Kesten [10]
and other authors on BBM with absorption.
2. Main results
2.1. Initial definitions
We consider a branching Brownian motion starting with one particle at the origin, whereby each particle moves independently and undergoes independent dyadic branching at exponential rater >0. We let the set of particles alive at timet beN (t ), and for each particleu∈N (t )denote its position at timet byXu(t). We extend this notion of a particle’s position to include the positions of its ancestors; that is, ifu∈N (t )has ancestorv∈N (s)for somes < t, then we setXu(s):=Xv(s). This setup will be given in more detail in Section4.
Fix a continuous functionf:[0,∞)→R, and anotherL:[0,∞)→(0,∞). Iff andLare twice continuously differentiable then we define
E(t ):=f(t)L(t )+ t
0
f(s)L(s)ds+1
2L(t)L(t )+1 2
t
0
L(s)L(s)ds
and
S:=lim inf
t→∞
1 t
t
0
r−1
2f(s)2− π2
8L(s)2+ L(s) 2L(s)
ds. (2.1)
We say that the pair(f, L)satisfies theusual conditionsif:
(I) f (0)=0;
(II) f andLare twice continuously differentiable;
(III) limt→∞E(t )/t=0;
(IV) S∈(−∞,∞).
We assume throughout this article that, unless otherwise stated, these conditions hold. We consider initially the class of sets of the form
B(f, L):=
g∈C[0,∞): g(t)−f (t )< L(t )∀t∈ [0,∞)
such thatf andLsatisfy the usual conditions. After we obtain our results we will be able to extend them in a natural way to cover more general subsets ofC[0,∞)– see Section7– but for now these conditions will allow us to apply integration by parts theorems without any complications. Although condition (III) may appear unnatural, there are clear reasons behind it, some of which are demonstrated via example in Section7. There are also similar conditions in the work on a single Brownian motion by Kesten [10] and Novikov [17].
Define N (t )ˆ :=
u∈N (t ): Xu(s)−f (s)< L(s)∀s≤t ,
the set of particles that have stayed within distance Lof the functionf for all times s≤t. We wish to study the number of particles inN (t )ˆ at large times. Let
Υ :=inf
t≥0: N (t )ˆ =∅ .
We callΥ theextinction timefor the process, and say that the process has becomeextinctby timet ifΥ ≤t. When we talk aboutsurvivalornon-extinction, we mean the eventΥ = ∞.
2.2. The non-critical case,S=0
We now state our main result in the non-critical case whenS=0. Most of this article will be concerned with proving this theorem.
Theorem 2.1. IfS <0,thenΥ <∞almost surely and logP(N (t )ˆ =∅)
infs≤t
s
0(r−f(u)2/2−π2/(8L(u)2)+L(u)/(2L(u)))du−→1.
On the other hand,ifS >0,thenP(Υ = ∞) >0and almost surely on survival we have log| ˆN (t )|
t
0(r−f(s)2/2−π2/(8L(s)2)+L(s)/(2L(s)))ds −→1.
As mentioned earlier, this theorem can be extended to cover more general sets, and we give results in this direction in Section7. The behaviour at criticality (S=0) depends on the finer behaviour off and L, but we are able to give some results in particular important cases in Section2.3below. We note the following corollary, which is easily deduced from Theorem2.1.
Corollary 2.2. IfS >0,then almost surely on survival we have lim sup
t→∞
1
t logN (t )ˆ =lim sup
t→∞
1 t
t
0
r− π2 8L(s)2−1
2f(s)2+ L(s) 2L(s)
ds and
lim inf
t→∞
1
t logN (t )ˆ =lim inf
t→∞
1 t
t
0
r− π2 8L(s)2−1
2f(s)2+ L(s) 2L(s)
ds.
This possibility of dramatic oscillation in the number of particles at large times is not usually seen in the branching processes literature. Example5, in Section3below, helps to show why it occurs in our situation.
2.3. The critical case,S=0
At least one obvious question immediately arises: what happens whenS=0? This is an interesting but delicate matter:
one must look at the finer behaviour of t
0
r−1
2f(s)2− π2
8L(s)2 + L(s) 2L(s)
ds.
Our methods, as they stand, are not always sharp enough to say what will happen, and we are unable to provide a complete theory as we must adapt carefully to the set in question. There are several situations, however, where something can be done. We are able to give results on the behaviour near the critical line√
2rt in Theorems2.3 and2.4below. Proofs of these two theorems will be given in Section8, as adaptations of our main proof, that of Theorem2.1.
Fixα >0,β∈(0,1)andγ >0, and fort≥0 let f (t )=α+√
2rt−α(t+1)β and L(t )=γ (t+1)β. Theorem 2.3. Ifβ <1/3then we haveP(Υ = ∞)=0,and
logP(N (t )ˆ =∅)
t1−2β −→ − π2
8γ2(1−2β).
Ifβ >1/3,we haveP(Υ = ∞) >0,and almost surely on survival log| ˆN (t )|
tβ −→(α+γ )√ 2r.
It is well-known that the asymptotic speed of the right-most particle in a BBM is√
2r. The theorem above concerns asking particles to stay close to this critical line forever: for example, we might ask particles to be in (√
2rt − 2αtβ,√
2rt )for all times t≥0. If β >1/3 then particles manage this with positive probability; ifβ <1/3 then they do not. What ifβ=1/3? Intuitively this question is “even more critical” than the previous theorem. Indeed, our methods are not able to give a full answer, but they can identify regimes where each behaviour (growth or death) is observed.
Theorem 2.4. Consider the caseβ=1/3.Let γ0:=
3π2 8√
2r 1/3
and γ1:=
3π2 4√
2r 1/3
.
Ifγ < γ0andα < 3π2
8γ2√
2r −γ,thenP(Υ = ∞)=0;in fact lim inf
t→∞
logP(N (t )ˆ =∅)
t1/3 ≥α√
2r−3π2 8γ2−γ√
2r and
lim sup
t→∞
logP(N (t )ˆ =∅)
t1/3 ≤α√
2r−3π2 8γ2 +γ√
2r.
On the other hand,ifγ≥γ1andα >3γ1/2,or ifγ < γ1andα > γ+8γ32π√22r,thenP(Υ = ∞) >0and almost surely on survival
lim inf
t→∞
log| ˆN (t )| t1/3 ≥α√
2r− 3π2
8(γ∨γ1)2−(γ∨γ1)√ 2r and
lim sup
t→∞
log| ˆN (t )| t1/3 ≤α√
2r−3π2 8γ2+γ√
2r.
Theorems2.3and2.4should be compared with what is currently known about the right-most particle, for example the work of Bramson [1] and Lalley and Sellke [12], results on branching Brownian motion with killing, for example Kesten [10], and work on the branching random walk, for example Hu and Shi [8] and Jaffuel [9]. The recent article by Jaffuel [9], in particular, gives results similar to our Theorems2.3and2.4.
3. Examples
We now consider some very simple examples to give the reader a flavour of the implications of Theorem2.1. More complex examples will be given in Sections7and8in order to explore the limits of our method.
Example 1. Takef (t )=λt withλ∈RandL(t )≡L >0.We have a growth rate ofr−λ22 −8Lπ22 (provided this is non-zero):if this constant is negative,then
1
t logP N (t )ˆ =∅
−→r−λ2 2 − π2
8L2
and if it is positive then there is a positive probability of survival,and almost surely on that event 1
t logN (t )ˆ −→r−λ2 2 − π2
8L2.
Thus taking a fixed L introduces an extra “killing” rate of 8Lπ22 to the system compared to the scaled results of [2,3,6,13].
Example 2. Again take f (t )=λt withλ∈R\ {√
2r}but now letLbe any unbounded monotone non-decreasing function such that(f, L)satisfies the usual conditions(for exampleL(t )=(t+1)β withβ∈(0,1)orL(t )=log(t+ 2)).Then we have a growth rate ofr−λ22:thus while constantLseverely restricts the growth of the system,as soon as we relaxLslightly we regain the full growth behaviour seen in[2,3,6,13].
Example 3. Letf (t )=√
2rt andL(t )≡L >0. Then we have extinction almost surely – and the same applies to anyf such thatt−1t
0f(s)2ds→2rwhen we take fixedL.We note that Theorems2.3and2.4provide much more interesting results in the same area.
Example 4. Letf (t )=λ(t+1)sin(log(t+1))andL(t )≡L.By simply working out the integrals in Corollary2.2 we see that ifris large enough then,on survival,the number of particles alive at timetoscillates with
lim inf
t→∞
1
t logN (t )ˆ =r− π2 8L2− λ2
√5 √
5+1 2
and
lim sup
t→∞
1
t logN (t )ˆ =r− π2 8L2− λ2
√5 √
5−1 2
. (Note the appearance of the golden ratio.)
The reason for this oscillation on the exponential scale becomes clearer when we consider the following simpler, but perhaps less natural, example.
Example 5. Define a continuous functionf:[0,∞)→Rby settingf (t )=0fort∈ [0,1]and f(t)=
0 if 22k≤t <22k+1 for some k∈ {0,1,2, . . .}, 1 if 22k+1≤t <22k+2 for some k∈ {0,1,2, . . .}.
Then,provided thatr >13+8Lπ22,on non-extinction we have lim inf
t→∞
1
t logN (t )ˆ =r− π2 8L2−1
3 and
lim sup
t→∞
1
t logN (t )ˆ =r− π2 8L2−1
6.
The idea here is that the number of particles grows quickly whenf(t)=0,but much more slowly whenf(t)=1as the steep gradient means that particles have to struggle to follow the path for a long time.As the size of the intervals [2n,2n+1]grows exponentially,the behaviour of the number of particles at timet is dominated by the behaviour on the most recent such interval. [We note that this choice off is not twice differentiable;however,it can be uniformly approximated by twice differentiable functions,and it is easily checked that our results still hold – see Section7.]
4. The spine setup
Consider a dyadic one-dimensional branching Brownian motion, branching at rate r, with associated probability measurePunder which:
• we begin with a root particle,∅, at 0;
• if a particleuis in the tree then all its ancestors are also in the tree (ifvis an ancestor ofuthen we writev < u);
• each particle uhas a lifetime σu, which is exponentially distributed with parameter r, and a fission time Su=
v≤uσv;
• each particleuhas a positionXu(t)∈Rat each timet∈ [Su−σu, Su);
• at the fission timeSu,uhas disappeared and been replaced by two childrenu0 andu1, which inherit the position of their parent;
• given its birth time and position, each particle u, while alive, moves according to a standard Brownian motion started fromXu(Su−σu)independently of all other particles.
For convenience, we extend the position of a particleuto all timest∈ [0, Su), to include the paths of all its ancestors:
Xu(t):=Xv(t) ifv≤uandSv−σv≤t < Sv.
We recall that we definedN (t )to be the set of particles alive at timet, N (t ):= {u: Su−σu≤t < Su},
and also that N (t )ˆ :=
u∈N (t ): Xu(s)−f (s)< L(s)∀s≤t .
We choose from our BBM one distinguished line of descent orspine– that is, a subset ξ of the tree such that ξ∩N (t )contains exactly one particle for eachtand ifu∈ξandv < uthenv∈ξ. We make this choice as follows:
• the initial particle∅is in the spine;
• at the fission time of nodeuin the spine, the new spine particle is chosen uniformly at random from the two children u0 andu1 ofu.
We denote the position of the spine particle at timet byξt; however we may also occasionally useξt to refer to the spine particle itself (that is, the node of the tree that is in the spine at timet) – it should be clear from the context which meaning is intended. We call the resulting probability measure (on the space ofmarked trees with spines)P.˜ We also consider the translated probability measuresPx andP˜x forx∈R, where underPx andP˜x we start with a single particle atxinstead of 0.
4.1. Filtrations
We use three different filtrations,Ft,F˜t andGt, to encapsulate different amounts of information. We give descriptions of these filtrations here, but the reader is referred to Hardy and Harris [4] for the full definitions.
• Ft contains all the information about the marked tree up to timet. However, it does not know which particle is the spine at any point.
• ˜Ft contains all the information about both the marked tree and the spine up to timet.
• Gt contains just the spatial information about the spine up to timet; it does not know anything about the rest of the tree.
We note thatFt⊆ ˜Ft andGt⊆ ˜Ft, and also thatP˜xis an extension ofPxin thatPx= ˜Px|F∞. 4.2. Martingales and a change of measure
UnderP˜, the path of the spine(ξt, t≥0)is a standard Brownian motion. Set G(t ):=exp
t 0
f(s)dξs−1 2
t 0
f(s)2ds+ t
0
π2 8L(s)2ds
·exp L(t)
2L(t ) ξt−f (t )2
− t
0
L(s)
2L(s) ξs−f (s)2
+ L(s) 2L(s)
ds
.
We claim that the process V (t ):=G(t )cos
π
2L(t ) ξt−f (t )
, t≥0 is aGt-local martingale.
Lemma 4.1. Let F (t ):=exp
t 0
π2
8L(s)2ds+ L(t) 2L(t )ξt2−
t 0
L(s)
2L(s)ξs2+ L(s) 2L(s)
ds
. The process
U (t):=F (t )cos πξt
2L(t )
is aGt-local martingale.
Proof. By Itô’s formula, dU (t )= π2
8L(t )2F (t )cos πξt
2L(t )
dt +
L(t)
2L(t )− L(t)2 2L(t )2
ξt2F (t )cos πξt
2L(t )
dt
− L(t)
2L(t )ξt2+ L(t) 2L(t )
F (t )cos πξt
2L(t )
dt +πL(t)
2L(t )2ξtF (t )sin πξt
2L(t )
dt +L(t)
L(t )ξtF (t )cos πξt
2L(t )
dξt
− π
2L(t )F (t )sin πξt
2L(t )
dξt
+ L(t)
2L(t )+ L(t)2 2L(t )2ξt2
F (t )cos
πξt 2L(t )
dt
− π2
8L(t )2F (t )cos πξt
2L(t )
dt
−πL(t)
2L(t )2ξtF (t )sin πξt
2L(t )
dt.
Lemma 4.2. The processV (t ),t≥0is aGt-local martingale.
Proof. Again applying Itô’s formula does the trick – or one may simply apply Girsanov’s theorem in series with
Lemma4.1.
By stopping the processV (t )at the first exit time of the spine particle from the tube{(x, t): |f (t )−x|< L(t )}, we obtain also that
ζ (t):=V (t )1{|f (s)−ξs|<L(s)∀s≤t}
is aGt-local martingale, and in fact since its size is constrained it is easily seen to be aGt-martingale. We call this martingaleζ thesingle-particle martingale.
Definition 4.1. We define anF˜t-adapted martingale by ζ (t)˜ =2n(ξ,t )×e−rt×ζ (t),
wheren(ξ, t ):= |{v: v < ξt}|is the generation of the spine at timet.The proof that this process is anF˜t-martingale can be found in[4].
We note that iff is anF˜t-measurable function then we can write:
f (t )=
u∈Nt
fu(t)1{ξt=u}, (4.1)
where eachfuisFt-measurable – intuitively,iff is in factGt-measurable,one replaces every appearance ofξt with Xu(t):so for example
Gu(t):=exp t
0
f(s)dXu(s)−1 2
t 0
f(s)2ds+ t
0
π2 8L(s)2ds
·exp L(t)
2L(t ) Xu(t)−f (t )2
− t
0
L(s)
2L(s) Xu(s)−f (s)2
+ L(s) 2L(s)
ds
.
It is also shown in[4]that if we define Z(t ):=
u∈N (t)
e−rtζu(t), (4.2)
whereζuis theFt-adapted process defined via the representation ofζ as in(4.1),then Z(t )= ˜Pζ (t)˜ |Ft
and hence thatZis anFt-martingale.This martingale is the main object of interest in this article.
Definition 4.2. We define a new measure,Q˜x,via dQ˜x
dP˜x
˜
Ft
=ζ (t)˜ ζ (0)˜ .
Also,for convenience,defineQxto be the projection of the measureQ˜ ontoF∞;then dQx
dPx
Ft =Z(t ) Z(0).
Lemma 4.3. UnderQ˜x,
• when at positionyat timet the spineξ moves as a Brownian motion with drift
f(t)+ y−f (t )L(t) L(t ) − π
2L(t )tan π
2L(t ) y−f (t )
;
• the fission times along the spine occur at an accelerated rate2r;
• at the fission time of nodevon the spine,the single spine particle is replaced by two children,and the new spine particle is chosen uniformly from the two children;
• the remaining child gives rise to an independent subtree,which is not part of the spine and which(along with its descendants)draws out a marked tree determined by an independent copy of the original measurePshifted to its position and time of birth.
This, again, was covered in [4]. We also use that, underQ˜x, the spine remains within distanceL(t )off (t )for all timest≥0. To see this explicitly, note that
Q˜x ξt∈ ˆ/N (t )
= ˜Px
1{ξ
t∈ ˆ/N (t)}
ζ (t)˜ ζ (0)˜
=0
by definition ofζ (t). All other particles, once born, move like independent standard Brownian motions but – as under˜ Px – we imagine them being “killed” instantly upon leaving the tube of radius Labout f. In reality they are still present in the system, but make no contribution toZonce they have left the tube.
Remark 4.1. Note thatNˆ,and henceZ,Q˜ and various other of our constructions,depend upon the choice of function f and radiusL.Usually these will be implicit,but occasionally we shall writeNˆf,L,Zf,LandQ˜f,L(and so on)to emphasise the choice off andLin use at the time.
4.3. Spine tools
We now state the spine decomposition theorem, which will be a vital tool in our investigation. It allows us to relate the growth of the whole process to just the behaviour along the spine. For a proof (of a more general version) the reader is again referred to [4].
Theorem 4.4 (Spine decomposition). We have the following decomposition ofZ:
Q˜x
Z(t )|G∞
= t
0
2re−rsζ (s)ds+e−rtζ (t).
The spine decomposition is usually used in conjunction with a result like the following – a proof of a more general form of this lemma can be found in [16].
Lemma 4.5. LetZ(∞)=lim supt→∞Z(t ).Then
QP ⇔ Z(∞) <∞ Q-a.s. ⇔ Q=Z(∞)P and
Q⊥P ⇔ Z(∞)= ∞ Q-a.s. ⇔ P Z(∞)
=0.
Another extremely useful spine tool is themany-to-onetheorem. A much more general version of this theorem is proved in [4], but the following version will be enough for our purposes.
Theorem 4.6 (Many-to-one). Iff (t )isGt-measurable for eacht≥0with representation(4.1),then P
u∈N (t)
fu(t)
=ertP˜ f (t )
.
We have one more lemma, a proof of which can be found in [7]. Although this result is extremely simple – and essential to our study – we are not aware of its presence in the literature before [7].
Lemma 4.7. For anyt∈ [0,∞](note that infinity is included here),we have Px Z(t ) >0
=Qx
Z(0) Z(t )
.
5. Almost sure growth along paths
5.1. Controlling the measure change
Before applying the tools that we have developed, we need the following short lemma to keep the Girsanov part of our change of measure under control.
Lemma 5.1. For anyu∈ ˆN (t ),almost surely under bothP˜xandQ˜xwe have t
0
f(s)dXu(s)− t
0
f(s)2ds
≤f(t)L(t )+f(0)x+ t
0
f(s)L(s)ds and hence underP˜
exp 1
2 t
0
f(s)2ds+ t
0
π2 8L(s)2ds−
t
0
L(s)
2L(s)ds−E(t )
≤Gu(t)≤exp 1
2 t
0
f(s)2ds+ t
0
π2 8L(s)2ds−
t 0
L(s)
2L(s)ds+E(t )
. (5.1)
Proof. From the integration by parts formula for Itô calculus, we know that f(t)Xu(t)=f(0)Xu(0)+
t 0
f(s)Xu(s)ds+ t
0
f(s)dXu(s).
From ordinary integration by parts, t
0
f(s)2ds=f(t)f (t)−f(0)f (0)− t
0
f (s)f(s)ds.
We also note that ifu∈ ˆN (t )then|Xu(s)−f (s)|< L(s)for alls≤t. Thus t
0
f(s)dXu(s)− t
0
f(s)2ds
=
f(t) Xu(t)−f (t )
−f(0) Xu(0)−f (0)
− t
0
f(s) Xu(s)−f (s) ds
≤f(t)L(t )+f(0)x+ t
0
|f(s)|L(s)ds.
Plugging this estimate into the definition ofGu(t)gives the result.
We are now ready to prove our first real result.
Proposition 5.2. Recall the definitions of S and Z from (2.1) and (4.2) respectively. Recall also that we set Z(∞):=lim supt→∞Z(t ).If S <0, then the process almost surely becomes extinct in finite time(and hence we haveZ(∞)=0).In this case,
logP(N (t )ˆ =∅) infs≤t
s
0(r−π2/(8L(u)2)−f(u)2/2+L(u)/(2L(u)))du−→1.
Alternatively,ifS >0thenP[Z(∞)] =1.
Proof. We first recall the spine decomposition and apply inequality (5.1):
Q˜
Z(t )|G∞
= t
0
2re−rsζ (s)ds+e−rtζ (t)
≤ t
0
2re−
s
0(r−π2/(8L(u)2)−f(u)2/2+L(u)/(2L(u)))du+E(s)ds +e−
t
0(r−π2/(8L(u)2)−f(u)2/2+L(u)/(2L(u)))du+E(t).
IfS >0, then the integrand above is exponentially small for all larget(as is the second term); so lim inft→∞Q[˜ Z(t )| G∞]<∞. It is easy to show that 1/Z is a positive(Q˜,Ft)-supermartingale, and henceZ(t )convergesQ-almost˜ surely to some (possibly infinite) limit. Thus, applying Fatou’s lemma, we get
Q˜
Z(∞)|G∞
≤lim inf
t→∞ Q˜
Z(t )|G∞
<∞.
We deduce thatZ(∞) <∞ ˜Q-almost surely, and Lemma4.5then gives thatP[Z(∞)] =1.
Alternatively, suppose thatS <0. Then by the above, Q˜
Z(t )|G∞
≤(2rt+1)e−infs≤t{
s
0(r−π2/(8L(u)2)−f(u)2/2+L(u)/(2L(u)))du−E(s)}. Now, by the tower property of conditional expectation and Jensen’s inequality,
P N (t )ˆ =∅
=P Z(t ) >0
=Q 1
Z(t )
≥ ˜Q 1
Q[˜ Z(t )|G∞]
.
This clearly implies that, for larget(using thatS <0), logP(N (t )ˆ =∅)
infs≤t
s
0(r−π2/(8L(u)2)−f(u)2/2+L(u)/(2L(u)))du
≤infs≤t{s
0(r−π2/(8L(u)2)−f(u)2/2+L(u)/(2L(u)))du−E(s)} −log(2rt+1) infs≤t
s
0(r−π2/(8L(u)2)−f(u)2/2+L(u)/(2L(u)))du ; and it is easy to see that the right-hand side converges to one ast→ ∞. This gives us the upper bound.
For the lower bound (still in the caseS <0), suppose for a moment that we may chooseγ >1 such that lim inf
t→∞
1 t
t 0
r−1
2f(s)2− π2
8γ L(s)2+ L(s) 2L(s)
ds <0.
We note that we may chooseγ in this way ift
0π2/8L(s)2ds(eventually) shows at most linear growth, which we will check later. Supposing this holds, then
P N (t )ˆ =∅
≤inf
s≤tP N (s)ˆ =∅
=inf
s≤tP
Zf,γ L(s)
Zf,γ L(s)1{ ˆNf,L(s)=∅}
=inf
s≤tQf,γ L 1
Zf,γ L(s)1{ ˆNf,L(s)=∅}
≤inf
s≤tQf,γ L
1{ ˆNf,L(s)=∅}
v∈ ˆNf,L(s)e−rsζvf,γ L(s)
.
IfNˆf,L(s)=∅then there is at least one particlevinNˆf,L(s); we may then apply inequality (5.1) toζvf,γ L(s)to see that
P N (t )ˆ =∅
≤inf
s≤t
1
e−0s(r−π2/(8γ2L(u)2)−f(u)2/2+L(u)/(2L(u)))du−γ2E(s)cos(π/2γ ) .
We repeat our calculations from the upper bound, taking logarithms and dividing by the desired denominator, to give logP(N (t )ˆ =∅)
infs≤t
s
0(r−π2/(8L(u)2)−f(u)2/2+L(u)/(2L(u)))du
≥infs≤t{s
0(r−π2/(8γ2L(u)2)−f(u)2/2+L(u)/(2L(u)))du−γ2E(s)} −log cos(π/2γ ) infs≤t
s
0(r−π2/(8L(u)2)−f(u)2/2+L(u)/(2L(u)))du
≥1+(1−1/γ2)sups≤ts
0π2/(8L(u)2)du+γ2sups≤tE(s)−log cos(π/2γ ) infs≤t
s
0(r−π2/(8L(u)2)−f(u)2/2+L(u)/(2L(u)))du (5.2) for larget. Thus it remains to check that the right-hand side above has a limsup that is close to 1 whenγ is close to 1.
Again it is sufficient thatt
0π2/8L(s)2dscan (eventually) show at most linear growth, and we check that fact now.
This is rather fiddly and not interesting in the context of the rest of the proof. Suppose it is not true; that is, suppose lim sup
t→∞
1 t
t
0
π2
8L(s)2ds= ∞. Then sinceS >−∞we must have
lim sup
t→∞
1 t
t
0
π2
8L(s)2 − L(s) 2L(s)
ds <∞. (5.3)
If we takeTn:=inf{t >0: t
0π2/8L(s)2ds > nt}, then d
dt 1
t t
0
π2 8L(s)2ds
Tn
>0, so differentiating and rearranging we get
L(Tn)2< π2Tn 8Tn
0 π2/(8L(s)2)ds <π2Tn 8n . Now, we note thatt
0 L(s)
L(s)ds=logL(t )−logL(0), so (5.3) implies that for all larget, t
0
π2
8L(s)2ds < Kt+1
2logL(t )
for some constantK. We have just shown thatL(Tn)2<π2Tn/8n, so for all largen, Tn
0
π2
8L(s)2ds < KTn+1
4logTn+1 4logπ2
8n contradicting (for largen) the definition ofTn.
We have shown that lim sup
t→∞
1 t
t 0
π2
8L(s)2ds <∞;
which allows us to make the limsup of (5.2) as close to 1 as we like by lettingγ↓1. This yields the lower bound, which in particular implies (by monotonicity) that the probability of eventual extinction is equal to 1.
5.2. Almost sure growth
Having established, in Proposition5.2, the large deviations behaviour of our model, we now turn to the question of what happens when extinction does not occur. The two propositions in this section contain the meat of our results in this direction. Proposition5.3gives a lower bound on the number of particles inN (t )ˆ for larget, and Proposition5.4 an upper bound. The former holds only on the event thatZhas a positive limit; as mentioned in the introduction, this set coincides (up to a null event) with the event that no particle manages to follow withinLoff, although we will not prove this fact until Section6. The proofs of our two propositions are very simple, but we stress again that this is due to the careful choice of martingale.
Proposition 5.3. LetΩbe the set on whichZhas a strictly positive limit, Ω:=
lim inf
t→∞ Z(t ) >0
.
IfS >0thenP-almost surely onΩwe have lim inf
t→∞
log| ˆN (t )| t
0(r−f(s)2/2−π2/(8L(s)2)+L(s)/(2L(s)))ds≥1.
Proof. For anyt≥0, by inequality (5.1), almost surely underP Z(t )=
u∈ ˆN (t )
e−rtζu(t)≤N (t )ˆ e−0t(r−π2/(8L(s)2)−f(s)2/2+L(s)/(2L(s)))ds+E(t). Hence (for larget, sinceS >0)
log| ˆN (t )| t
0(r−f(s)2/2−π2/(8L(s)2)+L(s)/(2L(s)))ds
≥logZ(t )+t
0(r−f(s)2/2−π2/(8L(s)2)+L(s)/(2L(s)))ds−E(t ) t
0(r−f(s)2/2−π2/(8L(s)2)+L(s)/(2L(s)))ds .
Now, onΩ we have lim inft→∞Z(t ) >0 and thus δt1 logZ(t )has a non-negative liminf for anyδ >0; then since
S >0 we see that the right-hand side above has liminf at least 1.
Remark 5.1. Recall that under P,Z is a non-negative martingale,and hence lim inft→∞Z(t )=Z(∞)P-almost surely.IfS >0,then by Proposition5.2P[Z(∞)] =1,so in this caseΩoccurs with strictly positive probability.
Proposition 5.4. IfS >0,thenP-almost surely we have lim sup
t→∞
log| ˆN (t )| t
0(r−f(s)2/2−π2/(8L(s)2)+L(s)/(2L(s)))ds≤1.
Proof. Fixγ >1 and letε=cos(π/2γ ). SinceZf,γ Lis a non-negative martingale underP, we haveZf,γ L(∞) <∞ P-almost surely. This implies that for anyδ >0, almost surely
lim sup
t→∞
1
δt logZf,γ L(t)≤0.
Now, almost surely underP,
Zf,γ L(t)=
u∈ ˆNf,γ L(t)
e−rtζuf,γ L(t)≥
u∈ ˆNf,L(t)
e−rtζuf,γ L(t).
By the definition ofεabove, for anyu∈ ˆNf,L(t)the cosine term inζuf,γ L(t)is at leastε(since the particle is within Loff (t )at timet). Applying inequality (5.1) we see that
Zf,γ L(t)≥Nˆf,L·ε·e−
t
0(r−π2/(8γ2L(s)2)−f(s)2/2+L(s)/(2L(s)))ds−γ2E(t)
and hence
log| ˆN (t )| t
0(r−f(s)2/2−π2/(8L(s)2)+L(s)/(2L(s)))ds
≤logZ(t )−logε+t
0(r−f(s)2/2−π2/(8γ2L(s)2)+L(s)/(2L(s)))ds+γ2E(t ) t
0(r−f(s)2/2−π2/(8L(s)2)+L(s)/(2L(s)))ds . As in Proposition5.2, we can bound the growth of thet
0 π2
8γ2L(s)2dsterm in the numerator so that lettingγ ↓1 we
get the desired result.
Corollary 5.5. IfS >0,thenP-almost surely on the eventΩ, log| ˆN (t )|
t
0(r−π2t/(8L2)−t
0f(s)2/2+L(s)/(2L(s)))ds−→1.
Proof. Simply combine Propositions5.3and5.4.
6. Showing thatZ(∞)=0 agrees with extinction
We note that we have now established our main result except for one key point: our growth results have so far been on the event{Z(∞) >0}, rather than the event of survival of the process,{Υ = ∞}. We turn now to showing that these two events differ only on a set of zero probability.
The approach to proving this is often analytic: one shows that P(Z(∞) >0)andP(Υ = ∞)satisfy the same differential equation with the same boundary conditions, and then shows that any such solution to the equation is unique. There is also sometimes a probabilistic approach to such arguments: one considers the product martingale
P (t ):=P Z(∞)=0|Ft
=
u∈N (t)
PXu(t) Z(∞)=0 .
On extinction, the limit of this process is clearly 1, and if we could show that on survival the limit is 0, then sinceP is a bounded non-negative martingale we would have
P(Υ <∞)=P P (∞)
=P P (0)
=P Z(∞)=0 .
In Harris et al. [5], for example, we have killing of particles at the origin rather than on the boundary of a tube – and it is shown that on survival, at least one particle escapes to infinity and its term in the product martingale tends to zero.
This is enough to complete the argument (although in [5] the authors favour the analytic approach). In our case we are hampered by the fact that for a single particleuthe value ofPXu(t)(Zu(∞)=0)is bounded away from zero, and if the particle is close to the edge of the tube, or even possibly in some places in the interior the tube, then this probability takes values arbitrarily close to 1.
The time-inhomogeneity of our problem means that other standard methods also fail. Our alternative approach is based upon similar principles as the probabilistic approach above, but is more direct: we show that if at least one particle survives for a long time, then it will have many births in “good” areas of the tube, and thusZ(∞) >0 with high probability.
Recall that underP˜x, we start at timet=0 with one particle at positionx(rather than at the origin) – and similarly for Q˜x. We assume throughout this section thatS >0, otherwise there is nothing to prove – our theorem does not
